1. The document analyzes the behavior of transmission probability in a single rectangular potential barrier where the barrier height and width are scaled by a common factor such that their product remains constant.
2. Expressions for transmission probability are derived for three cases: particle energy greater than, equal to, and less than the barrier energy. Approximations are used to express transmission probability in terms of the constant barrier height-width product.
3. The results show that transmission probability remains constant for particle energy greater than and less than the barrier energy. However, transmission probability decreases with decreasing particle energy when it is equal to the barrier energy.
1) The document analyzes the scattering properties of a conducting cylinder coated with an anisotropic chiral material using Mie's approach. Boundary conditions are used to derive a set of equations relating the scattered field coefficients, which are solved numerically.
2) Differential scattering cross sections are calculated for co-polarized and cross-polarized waves. Numerical results show the influence of chirality, permittivity, and permeability on the scattering response. Increasing chiral coating thickness decreases the co-polarized cross section.
3) Comparisons with previous work on a perfectly conducting chiral coated cylinder show good agreement. The scattering pattern has a maximum in the forward direction, which decreases with increasing chirality.
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
I am William T. I am a Solid Mechanics Assignment Expert at solidworksassignmenthelp.com. I hold a Bachlor's Degree in Engineering from McMaster University, Canada. I have been helping students with their Assignments for the past 10 years. I solve Assignments related to Solid Mechanics.
Visit solidworksassignmenthelp.com or email info@solidworksassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Solid Mechanics Assignments.
This document provides an introduction to the concepts covered in the course EC 8451 - Electromagnetic Fields. It begins with an overview of the electromagnetic model and defining the basic quantities used, including electric charge, current density, and the four fundamental field quantities. It then reviews key concepts in vector algebra and describes the rectangular, cylindrical, and spherical coordinate systems. The remainder of the document provides more details on units and constants, vector operations, and the Cartesian and cylindrical coordinate systems.
This document is a project report submitted by Shubham Patel for the partial fulfillment of an M.Sc. in Physics. The report introduces Galilean electromagnetism and constrained Hamiltonian systems. In part one, the report discusses various Galilean limits of Maxwell's equations including the electric limit, magnetic limit, and Carrollian limit. It also discusses formulations of these limits that are invariant under different systems of units. In part two, the report discusses Maxwell's field theory from a Hamiltonian perspective and constraints that arise in the formulation. It also discusses a higher order field tensor Lagrangian and its Hamiltonian formulation.
This document summarizes key concepts about electric potential and potential energy from Chapter 3:
1) Electric potential is defined as the work required per unit charge to move a test charge between two points in an electric field, similar to how gravitational potential is defined. 2) In a uniform electric field, the electric potential decreases as one moves in the direction of the field lines, corresponding to a decrease in potential energy for a positive charge. 3) The change in electric potential between two points depends only on the endpoints and not the path taken, as electric fields are conservative.
Effect of Piezoelectric Layer on Beam Parameters using Zigzag TheoryIDES Editor
An efficient higher order theory is presented for
static analysis of multilayered composite beams with
piezoelectric layers embedded or bonded to the surface, under
static electromechanical load. In this theory, the in-plane
displacement field is taken as a combination of a layer-wise
linear variation and a cubic variation across the thickness.
Transverse normal strains are neglected. The electric field is
also approximated as piecewise linear across the sub layers.
The displacement field is expressed in terms of only three
primary displacement variables excluding electric potential
variables by enforcing the conditions of zero transverse shear
stress at top and bottom of the beam and its continuity at layer
interfaces under general electromechanical loading. The effect
of thickness of the piezo-layer is observed for various loading
conditions. Also, the effect of beam lay-up on various
parameters is studied.
Electronic bands structure and gap in mid-infrared detector InAs/GaSb type II...IJERA Editor
We present here theoretical study of the electronic bands structure E (d1) of InAs (d1=25 Å)/GaSb (d2=25 Å) type
II superlattice at 4.2 K performed in the envelope function formalism. We study the effect of d1 and the offset ,
between heavy holes bands edges of InAs and GaSb, on the band gap Eg (), at the center of the first Brillouin
zone, and the semiconductor-to-semimetal transition. Eg (, T) decreases from 288.7 meV at 4.2 K to 230 meV
at 300K. In the investigated temperature range, the cut-off wavelength 4.3 m ≤ c ≤ 5.4 m situates this sample
as mid-wavelength infrared detector (MWIR). Our results are in good agreement with the experimental data
realized by C. Cervera et al.
1) The document analyzes the scattering properties of a conducting cylinder coated with an anisotropic chiral material using Mie's approach. Boundary conditions are used to derive a set of equations relating the scattered field coefficients, which are solved numerically.
2) Differential scattering cross sections are calculated for co-polarized and cross-polarized waves. Numerical results show the influence of chirality, permittivity, and permeability on the scattering response. Increasing chiral coating thickness decreases the co-polarized cross section.
3) Comparisons with previous work on a perfectly conducting chiral coated cylinder show good agreement. The scattering pattern has a maximum in the forward direction, which decreases with increasing chirality.
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
I am William T. I am a Solid Mechanics Assignment Expert at solidworksassignmenthelp.com. I hold a Bachlor's Degree in Engineering from McMaster University, Canada. I have been helping students with their Assignments for the past 10 years. I solve Assignments related to Solid Mechanics.
Visit solidworksassignmenthelp.com or email info@solidworksassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Solid Mechanics Assignments.
This document provides an introduction to the concepts covered in the course EC 8451 - Electromagnetic Fields. It begins with an overview of the electromagnetic model and defining the basic quantities used, including electric charge, current density, and the four fundamental field quantities. It then reviews key concepts in vector algebra and describes the rectangular, cylindrical, and spherical coordinate systems. The remainder of the document provides more details on units and constants, vector operations, and the Cartesian and cylindrical coordinate systems.
This document is a project report submitted by Shubham Patel for the partial fulfillment of an M.Sc. in Physics. The report introduces Galilean electromagnetism and constrained Hamiltonian systems. In part one, the report discusses various Galilean limits of Maxwell's equations including the electric limit, magnetic limit, and Carrollian limit. It also discusses formulations of these limits that are invariant under different systems of units. In part two, the report discusses Maxwell's field theory from a Hamiltonian perspective and constraints that arise in the formulation. It also discusses a higher order field tensor Lagrangian and its Hamiltonian formulation.
This document summarizes key concepts about electric potential and potential energy from Chapter 3:
1) Electric potential is defined as the work required per unit charge to move a test charge between two points in an electric field, similar to how gravitational potential is defined. 2) In a uniform electric field, the electric potential decreases as one moves in the direction of the field lines, corresponding to a decrease in potential energy for a positive charge. 3) The change in electric potential between two points depends only on the endpoints and not the path taken, as electric fields are conservative.
Effect of Piezoelectric Layer on Beam Parameters using Zigzag TheoryIDES Editor
An efficient higher order theory is presented for
static analysis of multilayered composite beams with
piezoelectric layers embedded or bonded to the surface, under
static electromechanical load. In this theory, the in-plane
displacement field is taken as a combination of a layer-wise
linear variation and a cubic variation across the thickness.
Transverse normal strains are neglected. The electric field is
also approximated as piecewise linear across the sub layers.
The displacement field is expressed in terms of only three
primary displacement variables excluding electric potential
variables by enforcing the conditions of zero transverse shear
stress at top and bottom of the beam and its continuity at layer
interfaces under general electromechanical loading. The effect
of thickness of the piezo-layer is observed for various loading
conditions. Also, the effect of beam lay-up on various
parameters is studied.
Electronic bands structure and gap in mid-infrared detector InAs/GaSb type II...IJERA Editor
We present here theoretical study of the electronic bands structure E (d1) of InAs (d1=25 Å)/GaSb (d2=25 Å) type
II superlattice at 4.2 K performed in the envelope function formalism. We study the effect of d1 and the offset ,
between heavy holes bands edges of InAs and GaSb, on the band gap Eg (), at the center of the first Brillouin
zone, and the semiconductor-to-semimetal transition. Eg (, T) decreases from 288.7 meV at 4.2 K to 230 meV
at 300K. In the investigated temperature range, the cut-off wavelength 4.3 m ≤ c ≤ 5.4 m situates this sample
as mid-wavelength infrared detector (MWIR). Our results are in good agreement with the experimental data
realized by C. Cervera et al.
Research on Transformer Core Vibration under DC Bias Based on Multi-field Cou...inventionjournals
The Mathematical models for DC bias vibration analysis of the transformer core are developed in this paper. The model is combined into multi-physical field coupling modeling for vibration analysis of the transformer. By applying the primary voltage as excitation and under different DC bias, vibrations of the transformer core is simulated and analyzed.
1) The document describes stress-strain diagrams from tensile tests on various materials including concrete, ceramics, steel, and alloys.
2) It provides data tables of load vs. strain measurements and asks the reader to plot stress-strain diagrams and determine values like modulus of elasticity, yield stress, and toughness.
3) Formulas are given for stress, strain, modulus of elasticity, and other mechanics of materials concepts as they relate to interpreting the stress-strain diagrams and tensile test data.
This document discusses vibration energy harvesting in uncertain environments. It begins with an introduction that describes piezoelectric vibration energy harvesting and the importance of considering uncertainty in both the input excitation and system parameters. It then presents single degree of freedom electromechanical models for linear and nonlinear energy harvesting systems. Optimal energy harvesting is analyzed for a linear system with a circuit without an inductor, where the mean harvested power is derived under Gaussian excitation.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
This document summarizes a new optical trapping technique based on optical binding forces. It involves using a single plane wave to trap particles by manipulating the scattering and binding forces between particles. The key points are:
1) Multiple small particles are arranged in fixed positions to collectively scatter and bind with an incident plane wave, producing oscillating binding forces.
2) Increasing the number of particles amplifies the binding forces, which can be tuned to cancel out the scattering force and produce a net negative force for trapping.
3) Optimizing the positions of 19 identical particles generates a stable two-dimensional trap, with minimal net force and a potential well to confine additional particles.
4) The technique
Application of particle swarm optimization to microwave tapered microstrip linescseij
Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering
because of their powerful features. In this work special design is done for a tapered transmission line used
for matching an arbitrary real load to a 50Ω line. The problem at hand is to match this arbitray load to 50
Ω line using three section tapered transmission line with impedances in decreasing order from the load. So
the problem becomes optimizing an equation with three unknowns with various conditions. The optimized
values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in
solving this kind of multiobjective optimization problems.
1) Fracture mechanics is the study of crack propagation in materials under stress. It considers failure as cracks growing through a structure rather than simultaneous failure.
2) Linear elastic fracture mechanics describes crack growth in elastic materials, while elastic-plastic fracture mechanics describes ductile crack growth in metals.
3) Fracture mechanics models failure using energy and strength criteria, considering the energy required to form new crack surface area.
This talk is about the analysis of nonlinear energy harvesters. A particular example of an inverted beam harvester proposed by our group has been discussed in details.
1. Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
2. The electric field intensity is defined as the force per unit charge. The electric field intensity due to a single point charge is calculated using Coulomb's law. For multiple point charges, the total electric field is calculated using the principle of superposition.
3. Continuous charge distributions can also produce electric fields. These include line charge distributions with linear charge density, surface charge distributions with surface charge density, and volume charge distributions with volume charge density. The electric field due to a uniform line charge distribution can be calculated by treating
1) The experiment involves dropping a magnet through a hollow conductive tube and analyzing the damped fall to study the tube's conductivity.
2) An equivalent circuit model is used to relate the damping coefficient to the tube's geometry, conductivity, and other parameters.
3) By equating the potential energy lost by the falling magnet to the energy dissipated in the tube via Joule heating, an expression is derived relating the induced electromotive force to the magnet's speed and the tube dimensions. This allows calculating the tube's conductivity.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
Magnetic hysteresis of soft magnetic material under an applied continuous ext...IAEME Publication
1. The document discusses modeling the hysteresis loops of soft magnetic materials like FeSi 3% that are subjected to an external magnetic field with both alternating current (AC) and direct current (DC) components.
2. It presents an approach using the Preisach model to predict asymmetric hysteresis loops under such combined fields. The Preisach model represents the magnetic state of each point in the material using elementary hysteresis operators.
3. A modified Lorentz distribution function is used to approximate the Preisach density function, allowing the model to account for parameters like coercivity and remanence that depend on the material and change with frequency.
This document discusses the Ewald summation method, which is a technique for efficiently calculating long-range electrostatic interactions in systems with periodic boundary conditions. It separates interactions into short-range and long-range parts for faster calculation. The method decomposes the lattice sum into real space and reciprocal space contributions. It also discusses applications of the Ewald summation method, including for uniformly charged surfaces, using fast Fourier transforms to improve computation time, and calculating interactions between molecules.
Comparison of Different Absorbing Boundary Conditions for GPR Simulation by t...IJMER
This paper compares three boundary conditions, i.e. transmitting boundary condition, Sarma
absorbing boundary condition and the uniaxial complete matched layerabsorbing boundary condition for
simulation of ground penetrating radar (GPR) by the time domain finite element (FEM) method. The
formulations of the three boundary conditions for the FEM method are described. Their effectiveness in
absorbing the incident electromagnetic waves are evaluated by the reflection coefficient on the boundary
of a simple GPR model.The results demonstrate that UPML boundary condition can yield a reflection
coefficient smaller than -50 dB, which is -20 dB smaller than other two boundary conditions.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Selection of Plastics by Design of Experimentstheijes
This work shows the optimization of two different types of plastic materials to measure a response variable, in order to select the best material that accomplishes the customer´s requirements in a company in the appliances sector. Factorial designs are an effective tool applied to compare two or more materials in order to choose the one that best accomplishes the requirement. The 24 factorial experimental design aims to study the effect of various factors on one or more response variables when it is necessary to know all factors. Factors considered in this case were: plasticity temperature, pigment, injection time and injection pressure. The objective of this work was to minimize weight as the response variable, this study was conducted between two plastics types: low-density polyethylene and nylon. The response variable was studied by an analysis of variance (ANOVA) and the equation of regression was obtained, to find the best array that optimizes both plastics separately with their different operating conditions and the customer can define which plastic is better taking decision based on the design of experiment as a powerful and effective tool that accomplishes the requirements. The results revealed that the most suitable plastic material is polyethylene because it fits with the specification that the client request.
The International Journal of Engineering and Sciencetheijes
1. The document summarizes observations from the CALLISTO spectrometer of type III and U bursts associated with an M7.9 solar flare on March 9th, 2012.
2. Analysis showed an inverted type U burst occurred between 310-384 MHz within 1 second, preceded by a group of type III bursts.
3. The bursts were associated with an M6 flare in Active Region 1429, and a halo coronal mass ejection was also observed.
The International Journal of Engineering and Sciencetheijes
This document presents a method for identifying weak nodes and branches in electric power systems using metric projections. Metric projections are applied to the Jacobian matrix from a state estimator to calculate distances between nodes. Results show the voltages at each node decrease as reactive power is increased at one node. Metric projections identify nodes 1, 3 and 4 as weak based on their distances from other nodes in the Jacobian matrix. This technique can help utilities improve reactive power support and transmission capacity by identifying stressed areas of the system.
Assessment of mortality and morbidity risks due to the consumption of some sa...theijes
This study assessed the mortality and morbidity risks from consuming sachet drinking waters produced in Abidjan, Côte d'Ivoire. Natural radioactivity measurements of 60 sachet water samples from 15 enterprises found concentrations of 40K, 226Ra, and 232Th ranging from 0.87-5.70 Bq/L, 0.16-0.47 Bq/L, and 0.17-0.60 Bq/L respectively. The estimated annual effective doses from ingesting these radionuclides were 45.48-113.07 μSv/y, with mortality and morbidity risks of 4.94×10-5 to 1.17×10-4 and
Research, Development Intelligent HVAC Control System Using Fuzzy Logic Contr...theijes
The paper describes an automatic climate in offices, describes the principles of the automation equipment climate, considered air parameters described control algorithms were compared automation system PIDcontroller and using fuzzy logic controller is designed microclimate model in Mathlab program with a fuzzy logic controller.
Organizational Structure and Software Project Success: Implications of the Me...theijes
This study examined the relationship between organizational structure (complexity, formalization, and centralization) and software project success, and whether organizational culture mediates this relationship. A survey was administered to 89 professionals from 10 banks to assess these variables. Results found the dimensions of organizational structure were significantly related to project success. However, organizational culture did not significantly mediate the relationship between organizational structure and project success. The study aims to contribute to understanding how organizational factors influence software project outcomes.
Research on Transformer Core Vibration under DC Bias Based on Multi-field Cou...inventionjournals
The Mathematical models for DC bias vibration analysis of the transformer core are developed in this paper. The model is combined into multi-physical field coupling modeling for vibration analysis of the transformer. By applying the primary voltage as excitation and under different DC bias, vibrations of the transformer core is simulated and analyzed.
1) The document describes stress-strain diagrams from tensile tests on various materials including concrete, ceramics, steel, and alloys.
2) It provides data tables of load vs. strain measurements and asks the reader to plot stress-strain diagrams and determine values like modulus of elasticity, yield stress, and toughness.
3) Formulas are given for stress, strain, modulus of elasticity, and other mechanics of materials concepts as they relate to interpreting the stress-strain diagrams and tensile test data.
This document discusses vibration energy harvesting in uncertain environments. It begins with an introduction that describes piezoelectric vibration energy harvesting and the importance of considering uncertainty in both the input excitation and system parameters. It then presents single degree of freedom electromechanical models for linear and nonlinear energy harvesting systems. Optimal energy harvesting is analyzed for a linear system with a circuit without an inductor, where the mean harvested power is derived under Gaussian excitation.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
This document summarizes a new optical trapping technique based on optical binding forces. It involves using a single plane wave to trap particles by manipulating the scattering and binding forces between particles. The key points are:
1) Multiple small particles are arranged in fixed positions to collectively scatter and bind with an incident plane wave, producing oscillating binding forces.
2) Increasing the number of particles amplifies the binding forces, which can be tuned to cancel out the scattering force and produce a net negative force for trapping.
3) Optimizing the positions of 19 identical particles generates a stable two-dimensional trap, with minimal net force and a potential well to confine additional particles.
4) The technique
Application of particle swarm optimization to microwave tapered microstrip linescseij
Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering
because of their powerful features. In this work special design is done for a tapered transmission line used
for matching an arbitrary real load to a 50Ω line. The problem at hand is to match this arbitray load to 50
Ω line using three section tapered transmission line with impedances in decreasing order from the load. So
the problem becomes optimizing an equation with three unknowns with various conditions. The optimized
values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in
solving this kind of multiobjective optimization problems.
1) Fracture mechanics is the study of crack propagation in materials under stress. It considers failure as cracks growing through a structure rather than simultaneous failure.
2) Linear elastic fracture mechanics describes crack growth in elastic materials, while elastic-plastic fracture mechanics describes ductile crack growth in metals.
3) Fracture mechanics models failure using energy and strength criteria, considering the energy required to form new crack surface area.
This talk is about the analysis of nonlinear energy harvesters. A particular example of an inverted beam harvester proposed by our group has been discussed in details.
1. Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
2. The electric field intensity is defined as the force per unit charge. The electric field intensity due to a single point charge is calculated using Coulomb's law. For multiple point charges, the total electric field is calculated using the principle of superposition.
3. Continuous charge distributions can also produce electric fields. These include line charge distributions with linear charge density, surface charge distributions with surface charge density, and volume charge distributions with volume charge density. The electric field due to a uniform line charge distribution can be calculated by treating
1) The experiment involves dropping a magnet through a hollow conductive tube and analyzing the damped fall to study the tube's conductivity.
2) An equivalent circuit model is used to relate the damping coefficient to the tube's geometry, conductivity, and other parameters.
3) By equating the potential energy lost by the falling magnet to the energy dissipated in the tube via Joule heating, an expression is derived relating the induced electromotive force to the magnet's speed and the tube dimensions. This allows calculating the tube's conductivity.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
Magnetic hysteresis of soft magnetic material under an applied continuous ext...IAEME Publication
1. The document discusses modeling the hysteresis loops of soft magnetic materials like FeSi 3% that are subjected to an external magnetic field with both alternating current (AC) and direct current (DC) components.
2. It presents an approach using the Preisach model to predict asymmetric hysteresis loops under such combined fields. The Preisach model represents the magnetic state of each point in the material using elementary hysteresis operators.
3. A modified Lorentz distribution function is used to approximate the Preisach density function, allowing the model to account for parameters like coercivity and remanence that depend on the material and change with frequency.
This document discusses the Ewald summation method, which is a technique for efficiently calculating long-range electrostatic interactions in systems with periodic boundary conditions. It separates interactions into short-range and long-range parts for faster calculation. The method decomposes the lattice sum into real space and reciprocal space contributions. It also discusses applications of the Ewald summation method, including for uniformly charged surfaces, using fast Fourier transforms to improve computation time, and calculating interactions between molecules.
Comparison of Different Absorbing Boundary Conditions for GPR Simulation by t...IJMER
This paper compares three boundary conditions, i.e. transmitting boundary condition, Sarma
absorbing boundary condition and the uniaxial complete matched layerabsorbing boundary condition for
simulation of ground penetrating radar (GPR) by the time domain finite element (FEM) method. The
formulations of the three boundary conditions for the FEM method are described. Their effectiveness in
absorbing the incident electromagnetic waves are evaluated by the reflection coefficient on the boundary
of a simple GPR model.The results demonstrate that UPML boundary condition can yield a reflection
coefficient smaller than -50 dB, which is -20 dB smaller than other two boundary conditions.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Selection of Plastics by Design of Experimentstheijes
This work shows the optimization of two different types of plastic materials to measure a response variable, in order to select the best material that accomplishes the customer´s requirements in a company in the appliances sector. Factorial designs are an effective tool applied to compare two or more materials in order to choose the one that best accomplishes the requirement. The 24 factorial experimental design aims to study the effect of various factors on one or more response variables when it is necessary to know all factors. Factors considered in this case were: plasticity temperature, pigment, injection time and injection pressure. The objective of this work was to minimize weight as the response variable, this study was conducted between two plastics types: low-density polyethylene and nylon. The response variable was studied by an analysis of variance (ANOVA) and the equation of regression was obtained, to find the best array that optimizes both plastics separately with their different operating conditions and the customer can define which plastic is better taking decision based on the design of experiment as a powerful and effective tool that accomplishes the requirements. The results revealed that the most suitable plastic material is polyethylene because it fits with the specification that the client request.
The International Journal of Engineering and Sciencetheijes
1. The document summarizes observations from the CALLISTO spectrometer of type III and U bursts associated with an M7.9 solar flare on March 9th, 2012.
2. Analysis showed an inverted type U burst occurred between 310-384 MHz within 1 second, preceded by a group of type III bursts.
3. The bursts were associated with an M6 flare in Active Region 1429, and a halo coronal mass ejection was also observed.
The International Journal of Engineering and Sciencetheijes
This document presents a method for identifying weak nodes and branches in electric power systems using metric projections. Metric projections are applied to the Jacobian matrix from a state estimator to calculate distances between nodes. Results show the voltages at each node decrease as reactive power is increased at one node. Metric projections identify nodes 1, 3 and 4 as weak based on their distances from other nodes in the Jacobian matrix. This technique can help utilities improve reactive power support and transmission capacity by identifying stressed areas of the system.
Assessment of mortality and morbidity risks due to the consumption of some sa...theijes
This study assessed the mortality and morbidity risks from consuming sachet drinking waters produced in Abidjan, Côte d'Ivoire. Natural radioactivity measurements of 60 sachet water samples from 15 enterprises found concentrations of 40K, 226Ra, and 232Th ranging from 0.87-5.70 Bq/L, 0.16-0.47 Bq/L, and 0.17-0.60 Bq/L respectively. The estimated annual effective doses from ingesting these radionuclides were 45.48-113.07 μSv/y, with mortality and morbidity risks of 4.94×10-5 to 1.17×10-4 and
Research, Development Intelligent HVAC Control System Using Fuzzy Logic Contr...theijes
The paper describes an automatic climate in offices, describes the principles of the automation equipment climate, considered air parameters described control algorithms were compared automation system PIDcontroller and using fuzzy logic controller is designed microclimate model in Mathlab program with a fuzzy logic controller.
Organizational Structure and Software Project Success: Implications of the Me...theijes
This study examined the relationship between organizational structure (complexity, formalization, and centralization) and software project success, and whether organizational culture mediates this relationship. A survey was administered to 89 professionals from 10 banks to assess these variables. Results found the dimensions of organizational structure were significantly related to project success. However, organizational culture did not significantly mediate the relationship between organizational structure and project success. The study aims to contribute to understanding how organizational factors influence software project outcomes.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Analysis of the skill of a world-class alpine ski racer by using a 3D CAD systemtheijes
Heavy snow areas account for about 50% of the total land area of Japan. Extension of healthy life expectancy is an important issue for residents of heavy snow areas because most of these areas in Japan are depopulated areas. Skiing is a possible lifelong sport in heavy snow areas because gravity rather than muscular power is mainly utilized in skiing. However, in Japan, the number of skiers has declined to 30% of the peak number. This study was conducted with the aim of recovery of the popularity of skiing, and establishing skiing as lifelong winter sport. World-level success of Japanese alpine ski racers is an important factor for the recovery of the popularity of skiing. In this study, the skill of an alpine ski racer was investigated by using a 3D CAD model of a skier with focus on joint work and energy balance in the turn motion.
Causes of Failure in Storage Facilities and Their Supportstheijes
Storage tanks in metallic or plastic forms are meant for storing water or other liquid substances for use either in industrial or domestic applications. This work looks into where failure occurs in metallic storage tanks, how there occur and why they occur. This study was important in other to proffer solutions and recommendations to the causes of failure recorded. The area of study was in Kaduna State, Nigeria, where four storage tanks were examined. The major causes of failure recorded were failure due to corrosion, failure due to malfunction of part, failure due to wrong construction methods among others. With corrosion of parts ranking highest on failure types, it was recommended that anti- corrosive materials like aluminum or stainless steel should be used for constructing storage tanks and their supports and when using mild steel materials it should be well protected using red oxide paint or hydrophobic polyester coatings.
Ethnobotanical documentation of some plants among Igala people of Kogi Statetheijes
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The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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The International Journal of Engineering and Science
1. The International Journal of Engineering
And Science (IJES)
||Volume|| 1 ||Issue|| 1 ||Pages|| 85-94 ||2012||
ISSN: 2319 – 1813 ISBN: 2319 – 1805
Behavior of Transmission Probability in a Single Rectangular
Potential Barrier at Constant Barrier Height–Barrier Width
Product
1,
Rupam Goswami, 2, Basab Das
1,2,
Department of Electronics & Communication Engineering, National Institute of Technology Silchar
---------------------------------------------------------- Abstract----------------------------------------------------------------
The application of Time Independent Schrodinger’s Wave Equation to understand the behavior of a particle in a
single rectangular potential barrier in one dimension has remained as one of the fundamental theories of
potential barrier problems in quantum physics. This work chiefly focuses on the transmission of a particle,
basically an electron, in such a barrier from a different perspective- behavior of transmission probability at
constant barrier height (energy)-barrier width product. Three basic cases of relationship between energy of the
particle and barrier potential energy have been discussed: energy of particle greater than barrier energy, energy
of particle equal to barrier energy and energy of particle less than barrier energy, where appropriate
approximations have been utilized to calculate the transmission probability in each case. The results have led to
major observations: if the energy of barrier is scaled by a constant and its width is scaled by the same constant
such that the barrier height-barrier width product always remains constant, then for a particular value of particle
energy, the transmission probability remains constant for the first and third case whereas in the second case, the
transmission probability reduces with decreasing energy and vice versa.
Keywords– Barrier height-barrier width product, rectangular potential barrier, scaling parameter,
Schrodinger’s Wave Equation, transmission probability
-------------------------------------------------------------------------------------------------------------------------------- --------
Date of Submission: 19, November, 2012 Date of Publication: 30, November 2012
----------------------------------------------------------------------------------------------------------------------------- ----------
1. INTRODUCTION
The rectangular potential barrier problem is a basic problem in modeling and studying various
semiconductor device interfaces (for example, oxide films). This work takes up a single rectangular potential
barrier in one dimension with a defined barrier height (energy) and barrier width. Thereafter, the barrier height
and barrier width are scaled to new values using a common scaling parameter such that their product always
remains same - this product has been referred to as barrier height-barrier width product during the course of the
paper. The purpose of this work is to express transmission probability of a particle (basically an electron) in
terms of the energy-width product of the potential barrier and study its variation with varying scaling parameter.
Although there are numerous models of single rectangular potential barrier where transmission probabilities of
particles are calculated and applied, the approach introduced in this work is another process of studying the
transmission probability in such barriers by identifying a new factor: barrier height-barrier width product. In this
work, the nature of the barrier height-barrier width product is at first defined. Using Time Independent
Schrodinger’s Wave Equation in one dimension to the single barrier introduced, the transmission probabilities
for three cases are derived: particle energy greater than barrier height, particle energy equal to barrier height and
particle energy less than barrier height. The expressions are then modified applying proper approximations
wherever required and calculations, to express transmission probabilities as functions of barrier height-barrier
width product. Necessary graphs are plotted to verify the analytical results.
2. PROPOSED PROCESS
A single rectangular barrier is taken into consideration as shown in Fig. 1, where the barrier height
(energy) is given by Vo and the barrier extends from x = 0 to x = a o. So, we have the barrier height- barrier width
product:
Now, if we divide the barrier height by a non-zero real number p so that it becomes V = V o/p and multiply the
barrier width by the same amount to get a = pao, then from (1) we have the barrier height-barrier width product:
www.theijes.com The IJES Page 85
2. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
From (2), it is seen that the barrier height-barrier width product remains constant if the above procedure of
scaling is used. This has been shown in Fig. 2.
Fig. 1: A simple rectangular potential barrier Fig. 2: Rectangular potential barrier
using defined scaling
The Time Independent Schrodinger’s Wave Equation (TISWE) in one dimension [1] for a particle is given by:
In equation (3), we have
Ѱ = Wave function
m = mass of the particle
E = total energy of the particle
V = scaled barrier height = Vo/p
Ћ = h/2π
The transmission probability of a particle in the rectangular barrier of Fig. 2 is found with the help of equation
(3) for three cases: E > V, E = V and E < V, where we consider the particle to be incident on the barrier from
left at x = 0, keeping the barrier height-barrier width product constant by using the scaling as discussed in the
preceding part. Thereafter we apply appropriate approximations wherever necessary to obtain the transmission
probabilities for the three cases in terms of barrier height-barrier width product.
3. EXPRESSIONS OF TRANSMISSION PROBABILITY
The single rectangular potential barrier [2] can be divided into three regions: Region I (x < 0), Region II (0 < x <
a) and Region III (x > 0) as shown in Fig. 3 and then TISWE is applied to each of the regions separately for the
three cases described.
Fig. 3: Three regions of the rectangular potential barrier to which TISWE is applied separately
www.theijes.com The IJES Page 86
3. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
The three cases under consideration are:
3.1 Case I: E > V
When the particle energy is lesser than the barrier energy, we have the solution of equation (3) in the
three regions as:
Here,
The solution of wave function given by expression (6) consists of only one term in the positive x-direction
because in Region III, there is no barrier present to reflect the particle.
The boundary conditions for this problem are:
Using one dimensional TISWE solutions (4), (5), (6) and boundary conditions (7), (8), (9), (10) we at first
express the constant A in terms of F. The transmission probability of a particle in this case will be given by its
relationship with the constant associated with the transmission of the particle in Region III and the constant
associated with incidence of the particle in Region I. Hence the transmission probability for E > V is:
3.2 Case II: E=V
For a particle incident on the barrier in Region I with energy equal to the barrier energy, the solution of
the TISWE in one dimension of equation (3) is given by:
The boundary conditions of this case are exactly similar to those of Case I and are given by boundary conditions
(7), (8), (9) and (10). Using solutions of one-dimensional TISWE (12), (13), (14) and following the same
procedure as in Case I, we get the transmission probability as:
3.3 Case III: E < V
In case of a particle with energy less than the barrier energy, the solutions of TISWE in one dimension
are given by:
where,
www.theijes.com The IJES Page 87
4. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
Using one-dimensional TISWE solutions (16), (17), (18) and boundary conditions (7), (8), (9), (10), and
following the same calculation steps as in Cases I and II, we get the transmission probability:
4. TRANSMISSION PROBABILITY (T) IN TERMS OF CONSTANT BARRIER HEIGHT-BARRIER
WIDTH PRODUCT (A)
It has been derived under proposed process that in a rectangular potential barrier, when the barrier
height (Vo) is reduced by ‘p’ times and the barrier width (ao) is increased by ‘p’ times, the barrier height-barrier
width product (A) is a constant independent of ‘p’. We can express transmission probability (T) of a particle in a
rectangular potential barrier in one dimension in terms of barrier height-barrier width product (A) for the three
cases under consideration using appropriate approximations in whichever case, applicable.
4.1 Case I: E > V
From equation (11), the transmission probability is:
Since a in equation (20) is very small, so, from Taylor’s series expansion of sin ( a), we can approximate:
Hence, using approximation (21) in equation (20), we get:
Substituting the values of and in equation (22), we get:
Therefore, transmission probability in terms of barrier height-barrier width product for E > V is:
4.2 Case II: E = V
Equation (15) gives:
www.theijes.com The IJES Page 88
5. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
So, transmission probability in terms of barrier height-barrier width product for E = V is:
4.3 Case III: E < V
From equation (19), the transmission probability is:
Since a is very small, so, from Taylor’s series expansion of sin ( a), we can approximate:
So, using approximation (25) in equation (19), we have:
Using the values of and in expression (26), we get:
www.theijes.com The IJES Page 89
6. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
So, transmission probability in terms of barrier height-barrier width product for E < V is:
5. RESULTS AND DISCUSSION
The results obtained during the calculation and approximations applied to the general expressions of
transmission probability for the three cases under consideration are discussed here.
5.1 Case I: E > V
5.1.1. Interpretation
Equation (23) shows the transmission probability (T) of a particle in terms of constant barrier height-
barrier width product (A) when the particle energy is greater than barrier height. In equation (23), the variables
m, A and are constants. So, T is a function of E only. Hence, observing the equation, we can conclude that:
For a particle with a particular value of energy E greater than scaled barrier height V, the transmission
probability is constant.
5.1.2. Table
In order to derive a simpler interpretation that transmission probability for Case I is constant for a particular E, it
is required to depict that the term in the denominator of expression (23) is a constant for a specific value of
E. Hence, equation (23) can be written as:
Table 1
E p L
10 eV 2 2.635 x 10-20
4
6
8
10
15 eV 3
4 1.757 x 10-20
7
8
10
20 eV 1 1.318 x 10-20
4
5
8
9
25 eV 2 1.054 x 10-20
3
6
9
10
For calculations in Table 1, the following values are considered:
Hence, barrier height-barrier width product, A =
www.theijes.com The IJES Page 90
7. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
We know that the barrier height or barrier width is varied by varying the scaling parameter p. From
equations (1) and (2), it is seen that the barrier height-barrier width is constant for any value of p. So, by plotting
a curve between L and scaling parameter p for different values of E greater than scaled barrier height V as listed
in Table 1, the result obtained in section 5.1.1 is proved.
Fig. 4: Plotof L versus p for Case I (E > V) with constant E
For a constant E, L is a constant. We have:
From the curve, L is a constant for a particular value of E. Hence from equation (29), T is constant for a
particular value of particle energy E.
5.2. Case II: E = V
5.2.1. Interpretation
Equation (24) expresses the transmission probability of Case II in terms of barrier height-barrier width product
when particle energy (E) is equal to scaled barrier height (V). Equation (24) is exactly similar to that of equation
(23). However, here since E = V = Vo/p, E is a function of scaling parameter p unlike in Case I and Case III.
Hence, there exists one curve between L and p for varying E. From equation (24), it is easily observed that with
increase in p, E (= V) decreases due to which the transmission probability reduces and vice-versa.
5.2.2. Table
As in the previous case we can modify equation (23) and write it as:
www.theijes.com The IJES Page 91
8. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
Table 2
p E = Vo/p L
2 25 eV 1.054 x 10-18
4 12.5 eV 2.108 x 10-18
6 8.33 eV 3.162 x 10-18
8 6.25 eV 4.216 x 10-18
10 5 eV 5.27 x 10-18
For calculations in Table 2, the following values are considered:
Hence, barrier height-barrier width product,
A=
5.2.3 Curve
The plot between L and scaling parameter p is shown in Fig. 4. It can be observed that with increase in p, L
increases. From equation (31), since T is inversely related to L, hence, T reduces with increasing p and vice-
versa- this establishes the interpretation in section 5.2.1.
Fig. 5: Plot of L versus p for Case II (E = V)
5.3. Case III: E < V
5.3.1. Interpretation
This case is exactly similar to that of Case I except that whenever particle energy is considered in curve
plots, it is kept below V. This is well depicted in the table and curve.
www.theijes.com The IJES Page 92
9. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
5.3.2. Table
Table 3
E p L
2 eV 2 13.18 x 10-18
4
6
8
10
4 eV 3 6.588 x 10-18
4
7
8
10
6 eV 1 4.392 x 10-18
4
5
8
9
8 eV 2 3.294 x 10-18
3
6
9
10
For calculations in Table 3, the following values are considered:
Hence, barrier height-barrier width product,
A=
5.3.3. Curve
Fig. 6: Plot of L versus p for Case I (E < V) with constant E
www.theijes.com The IJES Page 93
10. Behavior of Transmission Probability in a Single Rectangular Potential Barrier at Constant Barrier Height–
Barrier Width Product
6. CONCLUSION
This work clearly depicts how transmission probability in a single rectangular potential barrier can be
studied in terms of barrier height-barrier width product. A new parameter is introduced to project the variation
of transmission probabilities in such barriers, which is important keeping in view the highest chances of
exercising these relationships in understanding semiconductor device mechanisms. The advantage of this work
is the simplicity in approach and deriving the expressions to obtain important properties of rectangular potential
barriers. As in Cases I and III, the transmission probabilities remain constant for a constant barrier height-barrier
width product for a particular value of energy E. Such results may be used in modeling barriers in devices where
constant transmission probability is to be achieved by varying the barrier height and barrier width appropriately.
On the other hand, results of Case II show that the transmission probability reduces with increasing scaling
parameter, p when particle energy is equal to scaled barrier height V. Rectangular potential barriers find
applications in designing conduction models of polycrystalline Silicon [3] and gate oxide-buried oxide in
Silicon-on-insulator (SOI) [4]. The results obtained in this work may be applied on such barriers present in
devices.
ACKNOWLEDGEMENT
The authors would like to thank Mrs. Brinda Shome (Bhowmick), Assistant Professor, Department of
Electronics and Communication Engineering, National Institute of Technology Silchar for her support and
guidance.
REFERENCES
[1] David J. Griffiths, Time Independent Schrodinger Wave Equation, Introduction to Quantum Mechanics, 2
(New Jersey: Prentice Hall, 1994) 20-74.
[2] Donald A. Neaman, Introduction to Quantum Mechanics, Semiconductor Device and Physics: Basic
Principles, 2 (New York: McGraw Hill, 2003) 24-55.
[3] Dinesh Prasad Joshi and Ram Sahai Srivastava, A model of electrical conduction in polycrystalline silicon,
IEEE Transactions on Electron Devices, 7(31), 1984, 920-927.
[4] Stephen M. Ramey and David K. Ferry, Threshold voltage calculation in ultrathin-film SOI MOSFETs
using the effective potential, IEEE Transactions on Nanotechnology, 2(3), 2003, 121-125.
Authors:
Rupam Goswami,
Studying Master of Technology,
Department of Electronics & Communication Engineering,
National Institute of Technology Silchar, Silchar – 788010, Assam, India
Specialization: Microelectronics & VLSI Design.
Basab Das,
Studying Master of Technology,
Department of Electronics & Communication Engineering,
National Institute of Technology Silchar, Silchar – 788010, Assam, India
Specialization: Microelectronics & VLSI Design.
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